(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The earth can be regarded as a sphere of radius R,

the volume density of charge distributed as [tex] \rho = \frac{\rho_s r}{R} [/tex]

where the density is 0 at thcce centre and rises linearly with radius until it reaches ps at the earth's surface

i) prove that the total charge on the earth is [tex] \pi \rho_s R^3 [/tex]

ii) use gauss's law to find an expression for E at the point r outside the earth (r>R)

iii) use gauss's law to find an expression for E within the earths interior (r<R)

iv) write expressions for the x-, y- and z- components of E at a point outside the earth and calculate [tex] \bf{\nabla .E }

2. Relevant equations

Qenclosed = [tex] \int_v \rho d\tau. [/tex] (for a volume)

[tex] \oint_S \bf{E} . d\bf{a} = \frac{1}{\epsilon} Qenclosed [/tex]

3. The attempt at a solution

[tex] \int_v \rho \dtau [/tex]

substituting in the given rho,

[tex] \int \rho_s \frac{r}{R} \dtau[/tex]

the outer radius r is = R so it just becomes R/R =1,

[tex] \rho_s *1 \int \dtau [/tex]

this is where I integrate the dtau and it should give me the volume of a sphere

[tex] \rho_s \frac{4}{3}\pi r^3 [/tex]

this sort of looks like the answer i'm looking for

HOWEVER I DONT KNOW HOW TO GET RID OF THE 4/3 constant!!!

what have I done wrong?

i've tried heaps of other methods, but I'm always getting some type of constant out the front

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# Total electric charge on the earth,

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